The average price of three items of furniture is Rs. 15000. If their prices are in the ratio 3 : 5 : 7, what is the price of the cheapest item of furniture?

Introduction / Context:
This problem uses the concepts of average and ratio together. You are given the average price of three items and the ratio in which their individual prices are divided. The task is to find the actual price of the cheapest item. Such questions are common in aptitude tests related to commerce, profit and loss, and basic arithmetic reasoning.

Given Data / Assumptions:

There are three items of furniture.
The average price of the three items is Rs. 15000.
The prices are in the ratio 3 : 5 : 7.
We are required to find the price of the cheapest item, which corresponds to the smallest ratio term 3.

Concept / Approach:
First, from the average, we compute the total price of all three items. Then we use the ratio 3 : 5 : 7 to represent the three prices as 3k, 5k and 7k. The sum of these three ratio parts must equal the total price. So we solve for k and then multiply by the smallest term in the ratio to get the price of the cheapest item.

Step-by-Step Solution:
Step 1: Compute the total price of all three items. Total price = average price * number of items = 15000 * 3 = Rs. 45000. Step 2: Express the three prices using a common multiplier. Let the prices be 3k, 5k and 7k, where k is a positive constant. Step 3: Set up the equation using the total price. 3k + 5k + 7k = 45000. This gives 15k = 45000. Step 4: Solve for k. k = 45000 / 15 = 3000. Step 5: Compute the cheapest price. Cheapest item corresponds to 3k, so price = 3 * 3000 = Rs. 9000.

Verification / Alternative Check:
We can verify by computing all three prices: 3k = 9000, 5k = 15000 and 7k = 21000. Their sum is 9000 + 15000 + 21000 = 45000. Dividing this total by 3 gives an average of 15000, which matches the given average. This confirms that the distribution and the cheapest price are correct.

Why Other Options Are Wrong:
Option Rs. 6000 would correspond to 2k if k were 3000, but the ratio does not contain 2 as a term, so it is not valid.
Option Rs. 7000 does not fit the ratio pattern because 7000 is not equal to 3k, 5k or 7k for integer k that gives total 45000.
Option Rs. 8888 does not fit any sensible ratio here and does not give the correct total and average when combined with the other implied prices.

Common Pitfalls:
A common error is to directly divide the average by the smallest ratio term or to misinterpret average as relating to each ratio part rather than the total. Some students also forget to multiply the average by the number of items to get the total before applying the ratio, which leads to wrong equations and therefore incorrect answers.

Final Answer:
The price of the cheapest furniture item is Rs. 9000.